Morphological evolution in heteroepitaxial thin film structures at the nanoscale Mikhail A. Grekov 1, a, Sergey A. Kostyrko 1, b

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1 Defect and Diffuion Forum Vol. 364 (05) pp - (05) Tran Tech Publication, Switzerland doi:0.408/ Morphological evolution in heteroepitaxial thin film tructure at the nanocale Mihail A. Greov, a, Sergey A. Kotyro, b St. Peterburg State Univerity, Univeriteti pr. 35, St. Peterburg, 98504, Ruia a magreov@mail.ru, b ergey.otyro@gmail.com Keyword: thin film, urface diffuion, morphological intability, ize effect, urface elaticity. Abtract. The aim of thi tudy i to reolve the phenomenon of formation of meocopic tructure on the urface of heteroepitaxial thin film ytem due to urface diffuion by conidering the effect of both urface and interface tree. Elatic tre field caued by curved urface i olved by uing the contitutive equation of linear elaticity for the bul and urface phae. Baed on the method of uperpoition, a boundary perturbation technique, Gourat-Koloov complex potential and Muhelihvili repreentation, the boundary value problem i reduced to the ucceive olution of a ytem of ingular and hyperingular integral equation for any order of approximation. Thi olution and thermodynamic approach allow u to derive a governing equation which give the amplitude changing of a urface roughne with time. Introduction The development in micro- and nanoelectronic ha provoed great experimental, theoretical and computer imulation activity that ha allowed a better undertanding of the urface defect formation in olid thin film ytem caued by tre-driven ma tranport mechanim. It wa found that the competition between elatic train energy and urface energy can lead to the growth of urface profile perturbation. The evolution of the urface morphology i governed by the variation of chemical potential along the curved urface. The firt theoretical invetigation of the morphological intability in treed olid date bac to the wor of Aaro and Tiller []. They found that the flat urface i untable under inuoidal perturbation in hape if wavelength i greater than the ome critical value. The critical wavelength i proportional to the ratio of the contant urface energy to the local elatic energy. The rate at which thee untable hape perturbation grow i controlled by urface diffuion. The ame reult wa later independently received by Grinfeld [] and Srolovitz [3]. In tudie of Spencer et al. [4], Freund and Jondottir [5], tability analye have been extended for a trained film-ubtrate ytem. Grilhe conidered contant elatic tre (urface tenion) and found that condition for tability of the urface are enitive to the ign of applied tre [6]. Chiu and Gao conidered the evolution of cycloid type urface by obtaining an analytical olution [7]. Yang, Srolovitz [8] and Spencer, Meiron [9] tudied the nonlinear evolution of a tre-driven urface intability. Within their framewor, numerical imulation revealed rich dynamic of urface evolution. Wu et al., uing an experimentally utified univeral binding-energy-ditance curve, conidered urface deformation dependent on urface tre and urface energy [0]. The urface tability condition of a treed olid wa re-examined. Here we hould alo note the wor of Panat et al. [], where the growth of urface undulation controlled by volume and urface diffuion wa invetigated. And lat but not leat, Kim and Vlaa conducted a tudy dealing with the morphological intability of the multilayer thin film tructure []. In the pat decade, uing finite [3] and boundary element method [4] a well a different continuum baed model [ ], the effect of variou practically important phyical (e.g., film and ubtrate tiffne, urface tre, lattice and thermal mimatch) and geometrical parameter (e.g., film thicne, initial profile of undulation) on the morphological intability of film urface were thoroughly analyzed. However, there i a lac of undertanding of the influence of urface and interface phae elaticity on the urface morphology evolution. All right reerved. No part of content of thi paper may be reproduced or tranmitted in any form or by any mean without the written permiion of TTP, (ID: /0/5,3:0:43)

2 Defect and Diffuion Forum Vol Both experimental and theoretical reult demontrate that at the nanocale, due to the increaing elatic urface-to-volume ratio, urface effect become important in mechanical and phyical behavior of thin film tructure. To decribe uch ind of effect, Gurtin and Murdoch developed the urface elaticity theory by modelling the urface a two-dimenional membrane with different material propertie adhering to the underlying bul material [5, 6]. Cammarata and Sieradzi theoretically predicted effective elatic modulu enhancement/reduction of 0%, when the thicne of the film reduce to below 5 nm [7]. Fartah et al. confirmed thi reult by the experiment for everal multilayered metal film [8]. After that Miller and Shenoy developed an efficient approach to explain the ize dependence of the elatic rigiditie of nanoized tructural element uch a plate and beam [9]. In tudie of Sun, Zhang [0, ] and Zhou, Huang [] demontrated by atomitic imulation that the value of elatic contant of thin film are thicne dependent and approach the bul value a the film thicne increae. Finally, He et al. propoed a continuum urface elaticity model and analyzed the ize-dependent deformation of nanofilm [3]. Motivated by thee reult, we have developed a continuum model baed on a thermodynamic approach and urface elaticity theory capable of examining: the urface-interface tiffne effect on the tability of nanoized film tructure under mall perturbation of different hape; the ize-dependent tre concentration around a nanocale urface defect formed a a reult of the morphological intability. Surface diffuion i conidered a the dominant mechanim of ma tranport. The driving force for atomic migration i the gradient of chemical potential aociated with an elatic and urface energy. The elatic deformation i treated a quai-tatic tate and i decribed in the term of a unified boundary perturbation method that we developed for a number of D problem of thin film mechanic [4, 5]. Problem Formulation Conider an iotropic film-ubtrate tructure under plane train condition. The ubtrate i idealized a an elatic half-plane. And the film i modeled a a trip of thicne h 0. The perturbation of a free urface i decribed by an arbitrary periodic function g( x,t ) at any moment of time t 0 z : z z x i h g( x, t), g( x, t) A ( t)co x, () 0 n n where max A ( t) / ( t) t 0, n /. n n According to the theory of urface elaticity [5, 6], the urface and interface are defined a a pre-treed elatic membrane coherently bonded to the boundary of the bul material. The contitutive equation of linear elaticity for bul and urface in the cae of the plane train problem are conequently ( ), nn nn tt ( ), tt tt nn, z, nt nt ()

4 Recent Development of Diffuion Procee and their Application: Fluid, Heat and Ma tt 0 0 tt Figure. General problem formulation for a treed film-ubtrate compoite with lightly curved urface., z.

3 4 Recent Development of Diffuion Procee and their Application: Fluid, Heat and Ma tt 0 0 tt Figure. General problem formulation for a treed film-ubtrate compoite with lightly curved urface., z. (3) In Eq. 3, are the urface contant imilar to Lame contant, in Eq. ; 0 i the reidual urface tre. The condition at the curved urface and planar interface are decribed by generalized Young-Laplace equation which tae the following form in the cae of D problem: ( z ) T ( z ), z, (4) tt ( z) x ( z ) ( z ) ( z ) i, z, (5) d( ) here T ( ) ( ) i, nn i nt, lim ( z) ;, h are the local principal curvature h dx zz i0 and the metric coefficient, accordingly. Since we aumed that urface phae and the ubtrate are coherent, the diplacement are continuou at the boundary and the interface: lim u( z) u ( z ), u u iu, z, zz u( z ) u ( z ) u ( z ) 0, u ( z ) lim u( z), z. zz i0 (6) The condition at infinity are lim i 0, lim 0, lim, (7) x x x here i a rotation angle of a material particle. A common reaon for a longitudinal tre i difference in contraction between the film and ubtrate material a well a defect in the crytal tructure. The high level of intrinic (reidual tree developed during the film depoition) and extrinic tree (reult from mimatch in the coefficient of thermal expanion of different layer during ubequent thermal proceing) coupled with an undulated urface generate evere tre concentration which lead to urface elf-diffuion activation. The driving force i aociated with a competition between bul U and urface U energy.

Defect and Diffuion Forum Vol. 364 5 Following Freund [6], the local chemical potential for the urface can be defined a: ( z, t) U( z, t) ( x, t) U ( z, t), z.

4 Defect and Diffuion Forum Vol Following Freund [6], the local chemical potential for the urface can be defined a: ( z, t) U( z, t) ( x, t) U ( z, t), z. (8) The variation in chemical potential give a rie of atom along the urface: J DC. T b (9) In Eq. 8, Eq. 9 i the atomic volume, D i the urface elf-diffuivity, C i the number of diffuion atom per unit area, b i the Boltzmann contant, and T i abolute temperature. Conervation of ma at each point along the urface give the normal velocity: v n J. (0) Under Eq., Eq. 8 and Eq. 9, the urface evolution equation Eq. 0 tae the form: g( x, t) g( x, t) DC K ( U z, t) U ( z, t), K, t x x bt () while auming x, g( x, t), g( x, t) ( t) f ( x, t), max f ( x, t) t 0. x A mentioned above, the elatic deformation i treated a quai-tatic tate. In order to integrate the urface evolution equation, we olve the correponding mixed problem of plane elaticity for a doubly connected domain coniting of: the contitutive equation of linear elaticity Eq. for the bul i and Eq. 3 for urface i ; the tatic boundary condition Eq. 4 and Eq. 5 which are formulated in the form of generalized Young-Laplace equation; the continuity condition for the diplacement on the urface and interface phae Eq.6; the condition at infinity Eq. 7. Boundary perturbation method Following the uperpoition principle [4, 5], the olution of formulated boundary value problem, pecifically the urface tt ( z) and interface ( z) tre, the bul tre vector ( z) nn( z) i nt ( z), the diplacement vector u( z) u( z) iu( z), i preented a: G( z, ) G ( z, ) G ( z, ), z, () where the function G( z, ), G ( z, ), G ( z, ) are repectively equal to ( z, ) for and du / dz, du / dz, repectively; are Poion ratio and the hear modulu of the phae ( z, ), ( z, ), du / dz for 4 3,, ;, i the Kronecer delta.

5 6 Recent Development of Diffuion Procee and their Application: Fluid, Heat and Ma Figure. Superpoition principle. In Eq., ( z), u () z are the vector of tree and diplacement arien in a homogeneou half-plane with wavy boundary and elatic propertie of the layer under the action of unnown elf-balanced urface load pz ( ) and urface tre ( ) z ( z ) p( z ) T ( z ), z, z a/ p( z ) p( z a), p( t) dt 0, z a/ (3) while the rotation angle of the material particle and tree at infinity are equal to zero at infinity. (z), u (z) are the vector of tree and diplacement arien in the two-component plane D D ( D ) with elatic propertie of the phae, and unnown ump of traction and diplacement u at the rectilinear interface under condition Eq. 7 at infinity. Boundary condition at (Eq. 4 Eq. 6) and contitutive equation (Eq. and Eq. 3) lead to the ytem of the boundary equation for the unnown function pz ( ), ( z), ( z ): tt ( z ) ( z ) T ( z ), tt ( z ) p( z ) T ( z ), ( z ) ( z ), 0 0 tt (4) ( z ) i ( z ) ( z ), u ( z ) u ( z ), ( z ) ( z ), z. i i i i tt i 0 0 tt i In accordance with [7], the tre () z and diplacement u () z vector are related to Gourat Koloov complex potential () z, () z, z by the following equality i G ( z, ) ( z) ( z) ( z) ( z) z z ( z) e (5)

6 Defect and Diffuion Forum Vol where i the angle between the direction of the area element with the normal n and the real axi x of the complex variable z x ix. Uing perturbation technique, expand function () z, () z, pz ( ) and ( z) in power erie' of the mall parameter : ( z) ( z), ( z) ( z), m m ( m) ( m) m0 m! m0 m! m m p( z ) p ( z ), ( z) ( z ), ( m) ( m) m0 m! m0 m! (6) and boundary value of function, ( m) at and function p ( m), ( m) into Taylor' erie in ( m) the vicinity of the line Imz 0, conidering x a parameter l l i f ( z) l i f ( z) ( z ) ( x ), ( z ) ( x ), ( ) ( l) ( m) ( m) ( m) ( m) l0 l! l0 l! l l i f ( x) l i f ( x) p ( z ) p ( x ), ( z ) ( x ). ( ) ( l) ( m) ( m) ( m) ( m) l0 l! l0 l! (7) Subtituting Eq. 5 7 into the econd boundary condition from Eq. 4 and equating the coefficient of the power m ( m 0,, ), we obtain the following equence of equation for boundary value of function z, () () z ( ) ( ) ( ) ( ) ( ), (8) ( m) x ( m) x i tt( m) x p( m) x Fm x where ( z ) lim ( z) and zz i0 Denote function n a: ( z ) lim ( z). zz i0 ( m) ( z), Im z 0, m( z) ( m) ( z), Im z 0, (9) then the boundary value problem Eq. 8 i reduced to the Riemann-Hilbert problem the olution of which can be written a: i ( m) ( t) p( m) ( t) Fm () t m( z) dt dt dt. i t z i t z i t z (0) Here Fn () z are nown function that depend on previou approximation. Taing into account ump condition Eq. 4 on the interface, we derive two equation for the boundary value of function () z and () z

7 8 Recent Development of Diffuion Procee and their Application: Fluid, Heat and Ma m ( x ) m ( x ) m ( x ) m ( x ), i ( z) ( z), m, () ( u ) ( z), m, 3 4. Introduce auxiliary function () z and V() z ( z) ( z), Imz 0 ( z) ( z), Imz 0, ( z) ; V ( z) ( z) ( z), Imz 0 ( z) ( z), Imz 0 () then the boundary value problem Eq. are alo reduced to Riemann-Hilbert problem the olution of which, imilar to Eq. 0, we write a i () t ( t) ( u ) ( t) ( z) dt dt, V ( z) dt v i t z i t z i t z (3) where v, A a reult, the ytem of boundary condition Eq. 4 for m -order approximation tae the form of the ytem of integral equation in the unnown expanion coefficient p ( ) ( m) x, ( ) ( m) x, ( ) tt( m) x, ( ) ( m) x ( m) x X x t p( m) t dt X x t i ( m) t dt Ym x (4) ( ) (, ) ( ) (, ) ( ) ( ), ( x ) X ( x, t) p ( t) dt X ( x, t) p ( t) dt X ( x, t) i ( t) dt ( m) ( m) ( m) ( m) ( m) 3 ( m) m X ( x, t) i ( t) dt X ( x, t) ( t) dt Y ( x ) (5) p ( x ) i ( x ) X ( x, t) p ( t) dt X ( x, t) p ( t) dt ( m) ( m) 3 ( m) 3 ( m) X ( x, t) i ( t) dt X ( x, t) i ( t) dt 3 ( m) 3 ( m) 3 33 ( m) 34 ( m) tt( m) m X ( x, t) i ( t) dt X ( x, t) i ( t) dt i ( x ) Y ( x ) (6)

8 Defect and Diffuion Forum Vol tt( m) 3 ( m) ( m) 3 ( m) 3 ( m) 4 33 ( m) 34 ( m) m ( x ) X ( x, t) X ( x, t) p ( t) dt X ( x, t) p ( t) dt X ( x, t) X ( x, t) ( t) dt X ( x, t) ( t) dt X ( x, t) ( t) dt X ( x, t) ( t) dt Y ( x ) (7) The ernel X i and right-hand ide i Y m of Eq. 4 Eq. 7 are nown on each tep of approximation. The periodicity of a film urface mae it poible to olve thi problem in a form of Fourier erie ( m) x Amn En x p( m) x Amn En x n n ( ) ( ), ( ) ( ), 3 4 ( m) x Amn En x tt( m) x Am nen x n n ( ) ( ), ( ) ( ), (8) i n where A, ( ) bx mn C En x e, bn in /. On the bai of Cauchy-type integral the ytem of integral equation Eq. 4 Eq. 7 i reduced to the linear ytem of algebraic equation for the unnown coefficient A and their conugate. The right-hand ide of each equation i determined from the Fourier erie expanion of the nown -periodic function i mn / i i i i m mn n mn m n n / (9) Y B E ( x ), B / Y ( t) E ( t) dt Uing the developed method, a tre and train ditribution modified due to the wavine of the urface Eq. i obtained in the firt-order approximation,, tt tt(0) tt() tt tt(0) tt(),. i i(0) i() i i(0) i() (30) The evolution of the urface hape g( x, t ) can be tudied by calculating elatic train energy U and urface energy U along the wavy urface z : U ( z, t) tt ( z, t) tt ( z, t) tt(0) tt(0) ( t) tt() ( x ) tt(0) tt(0) tt() ( x ), U ( z, t) tt ( z, t) tt ( z, t) tt(0) tt(0) ( t) tt() ( x ) tt(0) tt(0) tt() ( x ). (3)

9 0 Recent Development of Diffuion Procee and their Application: Fluid, Heat and Ma After that, the urface evolution equation Eq. can be rewritten a: g( x, t) ( x, t) (, ) 4 tt() g x t K tt(0) 4 tt(0) tt(0) t x x (3) It follow that the olution of the partial differential equation Eq. 3, ubect to initial condition An(0) an, mut have the form of a Fourier erie repreentation Eq., indicating an exponential growth of all Fourier wavemode An () t ln Pn (, h0,,,,,,,,, 0, 0, K, ) t an (33) while cr, where critical wavelegth cr i determined from equation: P (, h,,,,,,,,,,, K, ) 0. (34) n Summary Baed on the Gibb thermodynamic and the urface continuum mechanic of Gurtin and Murdoch, we obtained a theoretical model that allow u: to predict nanocale pattern formation at thin film urface; to evaluate the effect of nanocale tructure on tre concentration near curved urface; to tudy ize effect in thin film tructure. The governing equation are two-dimenional and give u an amplitude change of each urface wavemode. In the nearet future, we are going to analyze: the influence of different hape perturbation on the tability of film urface; the dependence of the urface defect formation on geometric and phyical parameter; the tre ditribution along the curved urface. Acnowledgment The wor wa upported by the Ruian Foundation for Baic Reearch under grant Reference [] R. J. Aaro and W. A. Tiller: Metal. Tran. Vol. 3 (97), pp [] M. A. Grinfeld: Sov. Phy. Dolady. Vol. 3 (986), pp [3] D. J. Srolovitz: Acta Metall. Vol. 37 (989), pp [4] B. J. Spencer, P. W. Voorhee and S. H. Davi: Phy. Rev. Lett. Vol. 67 (99), pp [5] L. B. Freund and F. Jondottir: J. Mech. Phy. Solid. Vol. 4 (993), pp [6] J. Grilhe: Acta Metall. Mater. Vol. (993), pp [7] C.-h. Chiu and H. Gao: Int. J. Solid Struct. Vol. 30 (993), pp [8] W. H. Yang and D. J. Srolovitz: Phy. Rev. Lett. Vol. 7 (993), pp

10 Defect and Diffuion Forum Vol. 364 [9] B. J. Spencer and D. I. Meiron: Acta Metall. Mater. Vol. 4 (994), pp [0] C. H.Wu, J. Hu and C.-h. Chen: Acta Mater. Vol. 46 (998), pp [] R. Panat, K. J. Hia and D. G. Cahill: J. of Appl. Phy. Vol. 97 (005), pp. 7. [] J.-H. Kim and J. J. Vlaa: Int. J. Sol. Struct. Vol.44 (007), pp [3] J. H. Prevot, T. J. Baer, J. Liang and Z. Suo: Int. J. Sol. Struct. Vol. 38 (00), pp [4] Z. Liu and H.-H. Yu: Thin Solid Film. Vol. 53 (006), pp [5] M. E. Gurtin and A. Murdoch: Arch. Rat. Mech. An. Vol. 57 (975), pp [6] M. E. Gurtin and A. Murdoch: Int. J. Sol. Struct. Vol. 4 (978), pp [7] R.C. Cammarata and K. Sieradzi: Phy. Rev. Lett. Vol. 6 (989), pp [8] A. Fartah, E. E. Fullerton, I. K. Schuller, S. E. Bobbin, J. W. Wagner, R. C. Cammarata, S. Kumar and M. Grimditch: Phy. Rev. B. Vol. 44 (99), p [9] R. E. Miller and V. B. Shenoy: Nanotech. Vol. (000), p. 39. [0] C. T. Sun and H. T. Zhang: J. Appl. Phy. Vol. 93 (003), pp. 8. [] H. T. Zhang and C. T. Sun: AIAA J. Vol. 4 (004), pp [] L. G. Zhou and H. C. Huang: Appl. Phy. Lett. Vol. 84 (004), p [3] L. H. He, C. W. Lim and B. S. Wu: Int. J. Sol. Struct. Vol. 4 (004), pp [4] Yu. I. Viulina, M. A. Greov and S. A. Kotyro: Mech. Solid. Vol. 45 (00), pp [5] M. A. Greov and S. A. Kotyro: J. Appl. Math. Mech. Vol. 77 (03), pp [6] L. B. Freund: Int. J. Sol. Struct. Vol. 3 (995), pp [7] N. I. Muhelihvili: Some baic problem of the mathematical theory of elaticity (Noordhoff, Leiden 977).

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Online supplementary information Electronic Supplementary Material (ESI) for Soft Matter. Thi journal i The Royal Society of Chemitry 15 Online upplementary information Governing Equation For the vicou flow, we aume that the liquid thickne